基于关节极限的冗余机械臂混沌动力学及控制
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摘要
考虑关节位移极限指标,研究冗余机械臂系统动力学特性及延迟反馈法对系统混沌运动控制。以平面3自由度机械臂为研究对象,利用Jacobian矩阵伪逆法得到平面机械臂系统的动力学模型。基于关节位移极限指标,建立冗余机械臂系统的混沌动力学状态方程,采用龙格库塔法对模型求解,利用相图、Poincaré图及Lyapunov指数图等进行分析。结果表明基于关节位移极限的冗余机械臂的自运动表现出混沌现象。在此基础上,利用延迟反馈法进行混沌控制,得到在合适的扰动权重参数条件下,延迟反馈控制能使冗余机械臂的混沌运动稳定在周期轨道上,且发现在混沌吸引子内存在除1倍周期以外的2倍周期以及3倍周期窗口,通过选取不同的扰动权重能使系统稳定在不同的周期轨道上。
Considering the index of joint limit displacement,the dynamic characteristics of redundant manipulator system and the control to chaotic motion by delayed feedback method were studied. Taking the planar 3-DOF manipulator as the research object,the dynamic model of the planar manipulator system was deduced by the pseudoinverse of the Jacobian method. Based on the index of joint limit displacement,the dynamic equation of redundant manipulator system was established. The model was solved by Runge-Kutta method and dynamic characteristics were analyzed by using phase diagram,Poincaré diagram and Lyapunov exponent diagram et al.The result shows that the self-motion of redundant manipulator based on the joint displacement limit exhibits chaos phenomenon. On this basis,the chaos of the system was controlled by using the delayed feedback method. It is obtained that delayed feedback control can make the chaos motion of redundant manipulator in stable periodic orbits under the conditions of suitable disturbance weight parameter. In addition,it is found that it also has the windows of 2 times period and 3 times period as well as 1 times period in the chaotic attractor. By selecting different disturbance weight,the system can be stabilized on different periodic orbits.
Considering the index of joint limit displacement,the dynamic characteristics of redundant manipulator system and the control to chaotic motion by delayed feedback method were studied. Taking the planar 3-DOF manipulator as the research object,the dynamic model of the planar manipulator system was deduced by the pseudoinverse of the Jacobian method. Based on the index of joint limit displacement,the dynamic equation of redundant manipulator system was established. The model was solved by Runge-Kutta method and dynamic characteristics were analyzed by using phase diagram,Poincaré diagram and Lyapunov exponent diagram et al.The result shows that the self-motion of redundant manipulator based on the joint displacement limit exhibits chaos phenomenon. On this basis,the chaos of the system was controlled by using the delayed feedback method. It is obtained that delayed feedback control can make the chaos motion of redundant manipulator in stable periodic orbits under the conditions of suitable disturbance weight parameter. In addition,it is found that it also has the windows of 2 times period and 3 times period as well as 1 times period in the chaotic attractor. By selecting different disturbance weight,the system can be stabilized on different periodic orbits.
引文
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[2]孙奎,谢宗武,刘宏,等.梯度投影法多准则优化的应用研究[J].控制与决策,2007,22(12):1433-1437.SUN K,XIE Z W,LIU H,et al. Application of Gradient Projection Method Performance Criteria Optimizaion[J].Control and Decision,2007,22(12):1433-1437.
[3]VARGHESE M,FUEHS A,MUKUNDAN R. Zero Dynamics In Kinematically Redundant Robots[C]//Proceedings of the 2nd IEEE International Conference on Systems Engineering. Pittsburgh,1990:66-69.
[4]SHRINIVAS L,GHOSAL A. Possible Chaotic Motions in a Feedback Controlled 2R Robot[C]//Proceeding of the Internal Conference on Robot and Automation. Washington,1996:1241-1246.
[5] RAVISHANKAR A S,GHOSAL A. Nonlinear Dynamics and Chaotic Motions in Feedback-controlled two-and three-degree-of-freedom robots[J]. The International Journal of Robots Research,1999,18(1):93-108.
[6]张登材,李立.基于PD控制的平面3R刚性冗余度机器人中的混沌现象[J].机械科学与技术,2004,23(5):519-522.ZHANG D C,LI L. Chaotic Motion of a Planar Rigid 3R Redundant Robot with a PD Controller[J]. Mechanical Science and Technology,2004,23(5):519-522.
[7] OTT E,GREBOGI C,YORKE J A. Control Chaos[J].Physical Review Letters,1990,64(6):1196-1199.
[8]PYRAGAS K. Continuous Control of Chaos by Self-controlling Feedback[J]. Physical Letters A,1992,170(6):421-428.
[9] CHEN G,YU X H. On Time Delay Feedback Control of Chaotic System[J]. IEEE Transaction on Circuits and Systems-I,1999,46(6):767-772.
[10]YU X. Tracking Inherent Periodic Orbits in Chaotic Dynamic Systems via Adaptive Variable Structure Time-delayed Self Control[J]. IEEE Transaction on Circuits and Systems-I,1999,46(11):1408-1411.
[11]裴文江,黄俊,刘文波,等.自适应延迟反馈控制混沌[J].控制理论与应用,1999,16(2):297-300.PEI W J,HUANG J,LIU W B,et al. Controlling Chaos by Adaptive Delay Feedback[J]. Control Theory and Applications,1999,16(2):297-300.
[12]罗晓曙,方锦清,孔令江,等.一种新的基于系统变量延迟反馈的控制混沌方法[J].物理学报,2000,49(8):1423-1427.LUO X S,FANG J Q,KONG L J,et al. A New Method for Control Chaos Based on Delay Feeback of System Variables[J]. Acta Physica Sinica,2000,49(8):1423-1427.
[13] ZGHAL H,DUBEY R V,EULE J A. Efficient Gradient Projection Optimization for Manipulators with Multiple Degrees of Redundancy[C]//Proceedings of IEEE Inter-national Conference on Robotics and Automation. Cincinnati,1990:1006-1011.
[14]LIEGEOIS A. Automatic Supervisory Control of the Configuration and Behavior of Multibody Mechanisms[J]. IEEE Transactions on Systems Man and Cybernetics,1977,7(12):868-871.